Use matrix inversion to solve the given system of linear equations. (You previously solved this system using row reduction.)
x	 + 	2y			 = 	0
y	 − 	z	 = 	0
x	 + 	3y	 − 	2z	 = 	1
(x, y, z) =

We can solve the given system of linear equations using matrix inversion by following these steps:

1. **Write the system in matrix form**:  
   
   The system of equations is:

   \[
   \begin{aligned}
   x + 2y &= 0 \\
   y - z &= 0 \\
   x + 3y - 2z &= 1
   \end{aligned}
   \]

   In matrix form, this can be written as:

   \[
   A \mathbf{x} = \mathbf{b}
   \]

   where:
   \[
   A = \begin{pmatrix}
   1 & 2 & 0 \\
   0 & 1 & -1 \\
   1 & 3 & -2
   \end{pmatrix}, \quad
   \mathbf{x} = \begin{pmatrix}
   x \\
   y \\
   z
   \end{pmatrix}, \quad
   \mathbf{b} = \begin{pmatrix}
   0 \\
   0 \\
   1
   \end{pmatrix}
   \]

2. **Find the inverse of matrix A**:

   To solve for \(\mathbf{x}\), we need to find the inverse of matrix \(A\), denoted as \(A^{-1}\), and then use the relation:

   \[
   \mathbf{x} = A^{-1} \mathbf{b}
   \]

Let's compute the inverse of \(A\) and use it to find \(\mathbf{x}\).

I will calculate this now.The solution to the system of equations using matrix inversion is:

\[
(x, y, z) = (2, -1, -1)
\]

Would you like more details on the matrix inversion process or have any questions?

Here are five related questions:
1. How is the inverse of a matrix calculated?
2. What are the conditions for a matrix to be invertible?
3. How can matrix inversion be applied to larger systems of equations?
4. What is the geometric interpretation of solving systems using matrix inversion?
5. How does matrix inversion compare to row reduction in terms of efficiency?

**Tip**: Matrix inversion works best for small systems, but for larger systems, methods like LU decomposition or iterative approaches are often more efficient.

Use matrix inversion to solve the given system of linear equations: x + 2y = 0, y − z = 0, x + 3y − 2z = 1.

Learn how to solve the system of linear equations using matrix inversion. This problem covers key linear algebra concepts such as matrix form (A * x = b), matrix inversion, and solving systems of equations using the inverse of a matrix. Suitable for advanced high school and college-level students.

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